In this talk we will discuss old and not so old inequalities on the volume of the orthogonal projections (sometimes called local Loomis-Whitney type estimates). We will explore connections of those inequalities to inequalities for Mixed Volumes. We will also show the links between those inequalities and a number of interesting inequalities in Convex Geometry which are inspired by sumsets estimates in additive combinatorics and classical facts from the information theory. This is a joint work with Matthieu Fradelizi, Mokshay Madiman and Mathieu Meyer.